Andrews-Gordon type series for the level 5 and 7 standard modules of the affine Lie algebra A (2) 2
aa r X i v : . [ m a t h . R T ] J un ANDREWS-GORDON TYPE SERIES FOR THE LEVEL 5 AND 7STANDARD MODULES OF THE AFFINE LIE ALGEBRA A (2)2 MOTOKI TAKIGIKU AND SHUNSUKE TSUCHIOKA
Abstract.
We give Andrews-Gordon type series for the principal charactersof the level 5 and 7 standard modules of the affine Lie algebra A (2)2 . We alsogive conjectural series for some level 2 modules of A (2)13 . Introduction
In this paper, we use the q -Pochhammer symbol: for n ∈ N , m ∈ N ∪ {∞} ,( x ; q ) ∞ := Y i ≥ (1 − xq i ) , ( x ; q ) n := n − Y i =0 (1 − xq i ) , ( a , . . . , a k ; q ) m := ( a ; q ) m · · · ( a k ; q ) m . The Andrew-Gordon identities.
The
Rogers-Ramanujan identities X n ≥ q n ( q ; q ) n = 1( q, q ; q ) ∞ , X n ≥ q n + n ( q ; q ) n = 1( q , q ; q ) ∞ (1)was one of the motivations for inventing the vertex operators [15, §
14] in the theoryof affine Lie algebras (see [22]). It started from Lepowsky-Milne’s observation [23]: χ A (1)1 (2Λ + Λ ) = 1( q, q ; q ) ∞ , χ A (1)1 (3Λ ) = 1( q , q ; q ) ∞ . Here, χ A ( λ ) (called the principal chacater ) stands for the principally specializedcharacter of the vacuum space Ω( V ( λ )) [11, §
7] for the integrable highest weightmodule (a.k.a. the standard module) V ( λ ) associated with a dominant integralweight λ ∈ P + of the affine Lie algebra g ( A ). We obey the numbering of verticesof the affine Dynkin diagram A in [15, §
4] and duplicate A (1)1 , A (2)2 , A (2)odd as Figure1. The level of P i ∈ I d i Λ i ∈ P + is given by P i ∈ I ˇ a i d i , where the colabel ˇ a i isthe number written on the vertex α i in the figure. We can expand χ A ( λ ) into anexplicit infinite product via Lepowsky’s numerator formula (see [4]).After the success of vertex operator theoretic proofs of the Rogers-Ramanujanidentities [25, 26, 27], it has been expected that, for each A and λ , there should exist“Rogers-Ramanujan type identities” whose infinite products are given by χ A ( λ ).The Andrews-Gordon identities (Theorem 1.1) can be seen as an instance of thisexpectation because of an existence of a vertex operator theoretic proof for it [24].Note that the infinite product (the right hand side) in Theorem 1.1 is equal to
Date : Jun 4, 2020.2010
Mathematics Subject Classification.
Primary 11P84, Secondary 05E10.
Key words and phrases. integer partitions, Rogers-Ramanujan identities, affine Lie algebras,vertex operators, Andrews-Gordon identities, q -series, hypergeometric series. (1)1 1 ◦ α ⇔ ◦ α A (2)2 2 ◦ α ◦ α A (2)2 ℓ − ◦ α − ◦ α | ◦ α − ◦ α − · · · − ◦ α ℓ − ⇐ ◦ α ℓ Figure 1.
The affine Dynkin diagrams A (1)1 , A (2)2 , A (2)odd . χ A (1)1 ((2 k − i )Λ + ( i − ). This is the case of level 2 k − k = 2 and i = 2 ,
1. An even levelanalog is known as the Andrews-Bressoud identities (see [31, § Theorem 1.1 ([2]) . Let ≤ i ≤ k . Putting N j = n j + · · · + n k − , we have X n , ··· ,n k − ≥ q N + ··· + N k − + N i + ··· + N k − ( q ; q ) n · · · ( q ; q ) n k − = ( q i , q k +1 − i , q k +1 ; q k +1 ) ∞ ( q ; q ) ∞ . As do the Rogers-Ramanujan identities, Theorem 1.1 has an intepretation as apartition theorem [12]. A partition theorem is a statement of the form “For any n ≥
0, partitions ( λ , · · · , λ ℓ ) of n with condtion C are equinumerous to partitionsof n with condition D ”. Theorem 1.1 is equivalent to the partition theorem, where C : 1 ≤ ∀ j ≤ ℓ − k + 1 , λ j − λ j + k − ≥ |{ ≤ j ≤ ℓ | λ j = 1 }| < i , D : 1 ≤ ∀ j ≤ ℓ, λ j , ± i (mod 2 k + 1).1.2. The main theorems.
In this paper (see also [9, Conjecture 1.1]), “Andrews-Gordon type series” stands for an infinite sum of the form X i , ··· ,i s ≥ ( − P sℓ =1 L ℓ i ℓ q P sℓ =1 a ℓℓ ( ℓ ) + P ≤ j
Concerning level 5 standard modules for A (2)2 , we have X i,j,k ≥ ( − k q ( i ) +8 ( j ) +2 ( k ) +2 ij +2 ik +4 jk + i +5 j + k ( q ; q ) i ( q ; q ) j ( q ; q ) k = 1[ q , q , q , q ; q ] ∞ (= χ A (2)2 (5Λ )) , X i,j,k ≥ ( − k q ( i ) +8 ( j ) +2 ( k ) +2 ij +2 ik +4 jk + i +7 j +3 k ( q ; q ) i ( q ; q ) j ( q ; q ) k = 1[ q, q , q , q ; q ] ∞ (= χ A (2)2 (Λ + 2Λ )) . Theorem 1.3.
Concerning level 7 standard modules for A (2)2 , we have X i,j,k ≥ q ( i ) +8 ( j ) +10 ( k ) +2 ij +2 ik +8 jk + i +4 j +5 k ( q ; q ) i ( q ; q ) j ( q ; q ) k = 1[ q, q , q , q , q , q ; q ] ∞ (= χ A (2)2 (5Λ + Λ )) , X i,j,k ≥ q ( i ) +8 ( j ) +10 ( k ) +2 ij +2 ik +8 jk + i +8 j +9 k ( q ; q ) i ( q ; q ) j ( q ; q ) k = 1[ q, q , q , q , q , q ; q ] ∞ (= χ A (2)2 (Λ + 3Λ )) . heorem 1.4. Concerning level 7 standard modules for A (2)2 , we have X i,j,k,ℓ ≥ ( − k q ( i ) +2 ( j ) +2 ( k ) +8 ( ℓ ) + ij + ik +2 iℓ +4 jk +4 jℓ +4 kℓ + i +3 j + k +6 ℓ ( q ; q ) i ( q ; q ) j ( q ; q ) k ( q ; q ) l = 1[ q , q , q , q , q , q ; q ] ∞ (= χ A (2)2 (7Λ )) , X i,j,k,ℓ ≥ ( − k q ( i ) +2 ( j ) +2 ( k ) +8 ( ℓ ) + ij + ik +2 iℓ +4 jk +4 jℓ +4 kℓ + i +2 j +4 k +8 ℓ ( q ; q ) i ( q ; q ) j ( q ; q ) k ( q ; q ) l = 1[ q, q , q , q , q , q ; q ] ∞ (= χ A (2)2 (3Λ + 2Λ )) . Note that the level 5 module V (3Λ + Λ ) is missing in Theorem 1.2, but χ A (2)2 (3Λ + Λ ) = 1[ q, q , q , q ; q ] ∞ = 1( q ; q ) ∞ , can be written as a form of Andrews-Gordon type series (e.g. put x = q in § Comments on the proofs.
The following steps are common in proving thata multisum is equal to an infinite product.(S1) Reduce the multisum to a single sum.(S2) Search for lists of identities which deduce the desired result.The step (S1) for Theorem 1.2, 1.3, 1.4 uses a similar technique to the proof ofKanade-Russell’s conjectures of modulo 12 [18, I , I ] by Bringmann et.al. [5, § § Theorem 1.5 ([33, (39)=(83),(38)=(86),(99),(94)]) . X n ≥ q n ( q ; q ) n = χ A (2)2 (5Λ ) , X n ≥ q n +2 n ( q ; q ) n +1 = χ A (2)2 (Λ + 2Λ ) , X n ≥ q n + n ( q ; q ) n = χ A (2)2 (7Λ ) , X n ≥ q n + n ( q ; q ) n +1 = χ A (2)2 (3Λ + 2Λ ) . Toward A (2)2 analog of the Andrews-Gordon identities. Let us recallthe previous studies for A (2)2 . For the level 2 case, the principal characters areobtained by inflating q to q from the infinite products in (1). For the level 3 (resp.level 4) case, the vacuum spaces are studied in [6] (resp. in [29]), which resultedin conjectural partition theorems (see [31, Theorem 5.2, Theorem 5.3, Conjecture5.5, Conjecture 5.6, Conjecture 5.7]) that were later proved in [3, 7, 35] (resp. [34]).The Andrews-Gordon type series are known as [21, Corollary 18] (resp. [34]), whichare duplicated in Theorem 7.2 (resp. Remark 3.1). For the level 3 case, see also § .5. Toward A (2) odd , level 2 analog of the Andrews-Gordon identities. Re-cently, related to the expectation mentioned in § D (3)4 (see [31, § A (2)2 ℓ +1 (see [19, § ℓ = 2, Andrews-Gordon type series for χ A (2)5 (Λ + Λ ) , χ A (2)5 (Λ )are found as in [21, (21),(22)]. As shown by [16], they are related with theclassical G¨ollnitz partition theorems [31, Theorem 2.42, Theorem 2.43]. When ℓ = 3, χ A (2)7 (Λ + Λ ) , χ A (2)7 (Λ ) are the infinite products in (1). When ℓ =4, Bringmann et.al. [5, § § § χ A (2)9 (Λ + Λ ) , χ A (2)9 (Λ ) , χ A (2)9 (Λ ) which were conjectured in [19, (3.2),(3.4),(3.6)]together with interpretations as partition theorems. See also [30].Noting the principal characters for level 4 modules of A (2)2 coincide with someof those for level 2 of A (2)11 (see Remark 3.1), in Conjecture 6.1, Conjecture 6.2,Conjecture 6.3) we give conjectural Andrews-Gordon type series for χ A (2)13 (Λ + Λ ) , χ A (2)13 (Λ ) , χ A (2)13 (Λ ) , χ A (2)13 (Λ ) . It would be interesting if one can find a pattern in Andrews-Gordon type seriesobtained so far. For example, some of them for level 4 and 5 modules of A (2)2 (resp.level 2 of A (2)11 and A (2)13 ) share a recursive structure as in Remark 3.1 and Remark6.5 that we can also observe in the original (Theorem 1.1): namely, the Andrew-Gordon series for χ A (1)1 ((2( k − − i )Λ + ( i − ) is obtained by deleting n k − (or substituting n k − = 0) in that for χ A (1)1 ((2 k − i )Λ + ( i − ).We hope this paper contributes to the expectation mentioned in § Organization of the paper.
The paper is organized as follows. In §
2, we showsome auxilary summation formulas via Euler’s identities, the q -binomial theoremand the q -WZ method. Then, we prove Theorem 1.2, Theorem 1.3, Theorem 1.4in § § § § § A (2)13 (resp. A (2)2 ). Acknowledgments.
We thank S. Kanade and M. Russell for helpful discussions.This work was supported by the Research Institute for Mathematical Sciences,an International Joint Usage/Research Center located in Kyoto University and theTSUBAME3.0 supercomputer at Tokyo Institute of Technology. S.T. was supportedin part by JSPS Kakenhi Grants 17K14154, 20K03506 and by Leading Initiativefor Excellent Young Researchers, MEXT, Japan. M.T. was supported in part byStart-up research support from Okayama University.2.
Preparations
Recall the
Euler’s identities and the q -binomial theorem [13, (II.1),(II,2),(II.3)]. X n ≥ x n ( q ; q ) n ( A ) = 1( x ; q ) ∞ , X n ≥ q ( n ) x n ( q ; q ) n ( B ) = ( − x ; q ) ∞ , X n ≥ ( a ; q ) n ( q ; q ) n x n ( C ) = ( ax ; q ) ∞ ( x ; q ) ∞ . emma 2.1. For any M ∈ Z ≥ , we have (1) X i,j ≥ i + j = M q j ( q ; q ) i ( q ; q ) j = 1( q ; q ) M , (2) X i,j ≥ i + j = M ( − j q ( j ) + j ( q ; q ) i ( q ; q ) j = ( q ; q ) M ( q ; q ) M , (3) X i,j ≥ i + j +2 k = M ( − j q ij + j + k ( q ; q ) i ( q ; q ) j ( q ; q ) k = ( − q ; q ) [ M/ ( q ; q ) [ M/ , (4) X i,j ≥ i + j = M ( − j q ( j ) + ij +2 j ( q ; q ) i ( q ; q ) j = ( q ; q ) M ( q ; q ) M .Proof. By (A), we get (1) as follows. X i,j ≥ q j ( q ; q ) i ( q ; q ) j x i + j = 1( x ; q ) ∞ xq ; q ) ∞ = 1( x ; q ) ∞ = X M ≥ x M ( q ; q ) M . Similarly, we get (2) by (A), (B), (C) as follows. X i,j x i ( q ; q ) i ( − j q ( j ) + j x j ( q ; q ) j = 1( x ; q ) ∞ ( xq ; q ) ∞ = X M ≥ ( q ; q ) M ( q ; q ) M x M . To prove (3), we calculate the generating series of both sides by (A), (B). X i,j,k ≥ ( − j q ij + j + k ( q ; q ) i ( q ; q ) j ( q ; q ) k x i + j +2 k = (cid:18) X j ≥ (cid:0) ( − xq ) j ( q ; q ) j X i ≥ ( xq j ) i ( q ; q ) i (cid:1)(cid:19) · X k ≥ ( x q ) k ( q ; q ) k = (cid:18) X j ≥ ( − xq ) j ( q ; q ) j xq j ; q ) ∞ (cid:19) · x q ; q ) ∞ = 1( x ; q ) ∞ x q ; q ) ∞ X j ≥ ( x ; q ) j ( q ; q ) j ( − xq ) j . Thus, the generating series of the left hand side is equal to 1( x ; q ) ∞ x q ; q ) ∞ ( − x q ; q ) ∞ ( − xq ; q ) ∞ by (C). We easily see that it is equal to the generating series of the right hand side: X M ≥ ( − q ; q ) [ M/ ( q ; q ) [ M/ x M = X N ≥ ( − q ; q ) N ( q ; q ) N x N (1 + x ) = (1 + x ) ( − x q ; q ) ∞ ( x ; q ) ∞ . In the proof of (4), we promise 1 / ( q ; q ) n = 0 if n <
0. For
M, j ≥
0, we let f M,j := ( − j q ( j ) +( M − j ) j +2 j ( q ; q ) M ( q ; q ) M − j ( q ; q ) j ( q ; q ) M so that it suffices to prove (by the q -WZ method [17]) P j ≥ f M,j = 1 (note that f M,j = 0 when j > M ). The q -Zeilberger algorithm helps us finding an expression g M,j = ( − j (1 − q M +1 − j − q M +2 − j ) q ( j ) +( M − j ) j +2 j ( q ; q ) M (1 + q M +1 )(1 − q M +1 )( q ; q ) M − j +1 ( q ; q ) j − ( q ; q ) M for which we can verify that g M,j = 0 ⇒ ≤ j − ≤ M and f M +1 ,j − f M,j = g M,j +1 − g M,j for any
M, j ≥
0. This implies P j ≥ f M +1 ,j − P j ≥ f M,j = 0 forany M ≥
0. Now we only need to see P j ≥ f ,j = 1, which is obvious. (cid:3) . Proof of Theorem 1.2
Note that the Andrews-Gordon type series is of the form X i,j,k ≥ ( − k q ( i ) +8 ( j ) +2 ( k ) +2 ij +2 ik +4 jk + i +(5+2 a ) j +(1+2 a ) k ( q ; q ) i ( q ; q ) j ( q ; q ) k (2)for a = 0 ,
1. We rewrite the inner sum on j and k as X j,k ≥ ( − k q ( j ) +2 ( k ) +4 jk +(5+2 a +2 i ) j +(1+2 a +2 i ) k ( q ; q ) j ( q ; q ) k = X M ≥ X j,k ≥ j + k = M ( − j + M q ( j ) +2 jk +4 j +2 ( M ) +(2 a +2 i +1) M ( q ; q ) j ( q ; q ) k . By Lemma 2.1 (4) (with q replaced by q ), we see (2) is reduced to X i,M ≥ ( − M ( q ; q ) M ( q ; q ) i ( q ; q ) M q ( i ) + M +2 iM + i +2 aM . (3)With (B), the inner sum on i is rewritten as X i ≥ q ( i ) +(2 M +1) i ( q ; q ) i = ( − q M +1 ; q ) ∞ = ( − q ; q ) ∞ ( − q ; q ) M = 1( q ; q ) ∞ ( q ; q ) M ( q ; q ) M . Hence, by Lemma 2.1 (2), we see (3) · ( q ; q ) ∞ is equal to X M ≥ ( − M q M +2 aM ( q ; q ) M ( q ; q ) M = X m,n ≥ ( − m q ( m ) +4 ( n ) +2 mn +(1+2 a ) m +(2+2 a ) n ( q ; q ) m ( q ; q ) n . Finally we use (B) to rewrite the inner sum on m as X m ≥ ( − m q ( m ) +(1+2 a +2 n ) m ( q ; q ) m = ( q a +2 n ; q ) ∞ = ( q ; q ) ∞ ( q ; q ) n + a . Thus, we see (2) = (3) · ( q ; q ) ∞ ( q ; q ) ∞ is equal to X n ≥ q ( n ) +(2+2 a ) n ( q ; q ) n + a ( q ; q ) n = X n ≥ q n +2 an ( q ; q ) n + a , which proves Theorem 1.2 in virtue of Theorem 1.5. emark 3.1. In [34, Theorem 2.2] the following identities are shown: X i,k ≥ ( − k q ( i ) +2 ( k ) +2 ik + i + k ( q ; q ) i ( q ; q ) k = 1[ q , q , q ; q ] ∞ (= χ A (2)2 (4Λ ) = χ A (2)11 (Λ )) , (4) X i,k ≥ ( − k q ( i ) +2 ( k ) +2 ik + i +3 k ( q ; q ) i ( q ; q ) k = 1[ q, q , q ; q ] ∞ (= χ A (2)2 (2Λ + Λ ) = χ A (2)11 (Λ + Λ )) , (5) X i,k ≥ ( − k q ( i ) +2 ( k ) +2 ik +2 i +3 k ( q ; q ) i ( q ; q ) k = 1[ q , q , q ; q ] ∞ (= χ A (2)2 (2Λ ) = χ A (2)11 (Λ )) . (6) We remark that (4) and (5) coincide with double sums obtained by taking the “ j = 0 part” of the triple sums in Theorem 1.2. Proof of Theorem 1.3
Note that the Andrews-Gordon type series is of the form X i,j,k ≥ q ( i ) +8 ( j ) +10 ( k ) +2 ij +2 ik +8 jk + i +(4+4 a ) j +(5+4 a ) k ( q ; q ) i ( q ; q ) j ( q ; q ) k (7)for a = 0 ,
1. We rewrite the inner sum on j and k in (7) as X j,k ≥ q ( j ) +10 ( k ) +8 jk +(2 i +4+4 a ) j +(2 i +5+4 a ) k ( q ; q ) j ( q ; q ) k = X M ≥ X j,k ≥ j + k = M q ( k ) + k +8 ( M ) +(4+2 i +4 a ) M ( q ; q ) j ( q ; q ) k . By Lemma 2.1 (2) (with q replaced by − q ), we see (7) is reduced to X i,M ≥ q ( i ) + i +2 iM +4 M +4 aM ( − q ; q ) M ( q ; q ) i ( q ; q ) M . (8)With (B), the inner sum on i is rewritten as X i ≥ q ( i ) +(1+2 M ) i ( q ; q ) i = ( − q M ; q ) ∞ = ( − q ; q ) ∞ ( − q ; q ) M = 1( q ; q ) ∞ ( − q ; q ) M . By (1), we see (8) · ( q ; q ) ∞ is reduced to X M ≥ q M +4 aM ( − q ; q ) M ( q ; q ) M ( − q ; q ) M = X M ≥ q M +4 aM ( q ; q ) M = 1( q a , q − a ; q ) ∞ . This is equal to ( q ; q ) ∞ · χ A (2)2 ((5 − a )Λ + (1 + 2 a )Λ ) and proves Theorem 1.3. Remark 4.1.
In Slater’s list [33, (79)=(98),(96)] , there are identities whose infiniteproducts matches those in Theorem 1.3 (but we do not need them). X n ≥ q n ( q ; q ) n = χ A (2)2 (5Λ + Λ ) , X n ≥ q n +2 n ( q ; q ) n +1 = χ A (2)2 (Λ + 3Λ ) . . Proof of Theorem 1.4
Note that the Andrews-Gordon type series is of the form X i,j,k,ℓ ≥ ( − k +(1 − a )( j + k ) q ( i ) +2 ( j ) +2 ( k ) +8 ( ℓ ) + ij + ik +2 iℓ +4 jk +4 jℓ +4 kℓ + i +(1+ a ) j +(3+ a ) k +(6+2 a ) ℓ ( q ; q ) i ( q ; q ) j ( q ; q ) k ( q ; q ) ℓ (9)for a = 0 , j and k from the expression in Theorem 1.4 when a = 0).We rewrite the inner sum on j, k and ℓ in (9) as X j,k,ℓ ≥ ( − k +(1 − a )( j + k ) q ( j ) +2 ( k ) +8 ( ℓ ) +4( jk + jℓ + kℓ )+(1+ a + i ) j +(3+ a + i ) k +(6+2 a +2 i ) ℓ ( q ; q ) j ( q ; q ) k ( q ; q ) ℓ = X M ≥ X j,k,ℓ ≥ j + k +2 ℓ = M ( − k +(1 − a )( j + k +2 ℓ ) q jk +2 k +2 ℓ +2 ( j + k +2 ℓ ) +(1+ a + i )( j + k +2 ℓ ) ( q ; q ) j ( q ; q ) k ( q ; q ) ℓ By Lemma 2.1 (3) (with q replaced by q ), we see (9) is reduced to X i,M ≥ ( − (1 − a ) M q ( i ) + i +2 ( M ) +(1+ a + i ) M ( − q ; q ) [ M/ ( q ; q ) i ( q ; q ) [ M/ . (10)With (B), the inner sum on i is rewritten as X i ≥ q ( i ) +(1+ M ) i ( q ; q ) i = ( − q M ; q ) ∞ = ( − q ; q ) ∞ ( − q ; q ) M = 1( q ; q ) ∞ ( − q ; q ) M . Hence, (10) · ( q ; q ) ∞ is equal to X M ≥ ( − (1 − a ) M q ( M ) +(1+ a ) M ( − q ; q ) [ M/ ( − q ; q ) M ( q ; q ) [ M/ = X M ≥ ( − (1 − a ) M q ( M ) +(1+ a ) M ( − q ; − q ) M . (11)Further, by Lemma 2.1 (1) (with q replaced by − q ), we see (11) is equal to X m,n ≥ ( − (1 − a )( m + n )+ n q ( m + n ) +(1+ a )( m + n )+ n ( q ; q ) m ( q ; q ) n . (12)Finally, by (B), (12) is reduced to X n ≥ q ( n ) +2 n ( q ; q ) n X m ≥ ( − m q ( m ) +2 mn + m ( q ; q ) m = X n ≥ q ( n ) +2 n ( q ; q ) n ( q n +1 ; q ) ∞ ( a = 0) , X m ≥ q ( m ) +2 m ( q ; q ) m X n ≥ ( − n q ( n ) +2 mn +3 n ( q ; q ) n = X m ≥ q ( m ) +2 m ( q ; q ) m ( q m +3 ; q ) ∞ ( a = 1) . In each case, we see (9) = (12) / ( q ; q ) ∞ is equal to X s ≥ q ( s ) +2 s ( q ; q ) s ( q s +1+2 a ; q ) ∞ ( q ; q ) ∞ = X s ≥ q ( s ) +2 s ( q ; q ) s ( q ; q ) s + a = X s ≥ q s + s ( q ; q ) s + a . This proves Theorem 1.4 in virtue of Theorem 1.5. . Conjectures for level 2 modules of A (2)13 The level 2 principal characters of A (2)13 are χ A (2)13 (( δ i + δ i )Λ + Λ i ) = ( q ; q ) ∞ ( q ; q ) ∞ [ q i ; q ] ∞ [ q i ; q ] ∞ , where 0 ≤ i ≤ δ is the Kronecker delta.We give conjectural Andrews-Gordon type series for χ A (2)13 (Λ +Λ ) and χ A (2)13 (Λ n +1 ),where n = 1 , ,
3. Note that χ A (2)13 (2Λ ) = χ A (2)13 (2Λ ) and χ A (2)13 (Λ n ), where n = 1 , ,
3, are obtained by inflating q to q from infinite products with smallerperiod. Conjecture 6.1.
We have F (2 , ,
2) = χ A (2)13 (Λ ) , F (4 , ,
6) = χ A (2)13 (Λ ) and F (6 , ,
6) = χ A (2)13 (Λ ) , where F ( a, b, c ) := X i,j,k ≥ ( − k q ( i ) +2 ( j ) +4 ( k ) +2 ij +4 ik +4 jk + ai + bj + ck ( q ; q ) i ( q ; q ) j ( q ; q ) k . Conjecture 6.2.
We have F (1 , ,
12) = χ A (2)13 (Λ + Λ ) , F (1 , ,
8) = χ A (2)13 (Λ ) and F (3 , ,
16) = χ A (2)13 (Λ ) , where F ( a, b, c ) := X i,j,k ≥ ( − j q ( i ) +2 ( j ) +16 ( k ) +2 ij +4 ik +4 jk + ai + bj + ck ( q ; q ) i ( q ; q ) j ( q ; q ) k . (13) Conjecture 6.3.
We have F (1 , , ,
12) = χ A (2)13 (Λ ) , where F ( a, b, c, d ) := X i,j,k,ℓ ≥ ( − k q ( i ) +4 ( j ) +2 ( k ) +16 ( ℓ ) +2 ij +2 ik +4 iℓ +4 jk +8 jℓ +4 kℓ + ai + bj + ck + dℓ ( q ; q ) i ( q ; q ) j ( q ; q ) k ( q ; q ) l . Remark 6.4.
One can prove F ( a, b, c ) = F ( a, a + 1 , b, c ) for a, b, c ≥ byrewriting the inner sum on i in (13) as X i ≥ q ( i ) +(2 j +4 k + a ) i ( q ; q ) i = X i ≥ X s,t ≥ s +2 t = i q ( s +2 t ) +(2 j +4 k + a )( s +2 t )+ ( s )( q ; q ) s ( q ; q ) t = X s,t ≥ q ( s ) +4 ( t ) +2 st +(2 j +4 k )( s +2 t )+ as +(2 a +1) t ( q ; q ) s ( q ; q ) t . Here, the first equality follows from X i,j ≥ i +2 j = M q ( i )( q ; q ) i ( q ; q ) j = 1( q ; q ) M for M ≥ ,which is proved similarly to Lemma 2.1 (1) (using (A) and (B)). Hence, if Conjecture 6.2 is true, we have F (1 , , ,
12) = χ A (2)13 (Λ +Λ ) , F (1 , , ,
8) = χ A (2)13 (Λ ), F (3 , , ,
16) = χ A (2)13 (Λ ) and Conjecture 6.3 gives the “missing” case. Remark 6.5.
The double sums (4) and (5) coincide with those obtained by takingthe “ k = 0 part” of the triple sums F (1 , , and F (1 , , in Conjecture 6.2. . Notes on Capparelli’s identities
We fix the conditions (C1) and (C2) on a partition λ = ( λ , · · · , λ ℓ ) to recallCapparelli’s partition theorems (Theorem 7.1). See also § (C1) ≤ ∀ j ≤ ℓ − λ j − λ j +1 ≥ (C2) ≤ ∀ j ≤ ℓ − λ j − λ j +1 ≤ ⇒ λ j + λ j +1 ≡ Theorem 7.1 ([3, 7, 35]) . Let a = 1 , . For any n ≥ , partitions λ of n withcondtion C a are equinumerous to those with condition D a , where C a : (C1), (C2) and ≤ ∀ j ≤ ℓ, λ i = a , D a : 1 ≤ ∀ j ≤ ℓ, λ j
6≡ ± a (mod 6) , and λ , . . . , λ ℓ ( λ ) are distinct. In [21, Theorem 10, Theorem 11], Kur¸sung¨oz showed f ( x, q ) = X i,j ≥ q ( i ) +12 ( j ) +6 ij +2 i +6 j ( q ; q ) i ( q ; q ) j x i +2 j , (14) f ( x, q ) = X i,j ≥ q ( i ) +12 ( j ) +6 ij +3 i +9 j ( q ; q ) i ( q ; q ) j x i +2 j (1 + xq i +3 j ) , (15)where C a denote the set of partitions with the condition C a and f a ( x, q ) := P λ ∈C a x ℓ ( λ ) q | λ | for a = 1 ,
2. Combining Theorem 7.1 and (14),(15) with x = 1, Kur¸sung¨oz got thefollowing identities. Theorem 7.2 ([21, Corollary 18]) . Concerning the level 3 modules of A (2)2 , we have X i,j ≥ q i +6 ij +6 j ( q ; q ) i ( q ; q ) j = ( − q , − q , − q , − q ; q ) ∞ (cid:18) = 1[ q , q ; q ] ∞ = χ A (2)2 (3Λ ) (cid:19) , X i,j ≥ q i +6 ij +6 j + i +3 j (1 + q i +3 j +1 )( q ; q ) i ( q ; q ) j = ( − q, − q , − q , − q ; q ) ∞ (cid:18) = [ q ; q ] ∞ [ q, q , q ; q ] ∞ = χ A (2)2 (Λ + Λ ) (cid:19) . Note that the left hand side of the latter is not an Andrews-Gordon type seriesin our sense (see § X i,j,k ≥ q ( i ) +5 ( j ) +12 ( k ) +3 ij +6 ik +6 jk +(3 − a ) i +(2+ a ) j +6 k ( q ; q ) i ( q ; q ) j ( q ; q ) k = χ A (2)2 ((5 − a )Λ + ( a − ) , for a = 1 ,
2. This follows from substituting x = 1 to Theorem 7.3. Theorem 7.3.
For a = 1 , , we have f a ( x, q ) = X i,j,k ≥ q ( i ) +5 ( j ) +12 ( k ) +3 ij +6 ik +6 jk +(3 − a ) i +(2+ a ) j +6 k ( q ; q ) i ( q ; q ) j ( q ; q ) k x i + j +2 k . Proof.
Since the set of partitions with the conditions (C1) , (C2) is a linked partitionideal (see [1, § q -difference equation algorithmically F ( x ) = (1 + xq ) F ( xq ) + x ( q − a + q a + xq ) F ( xq ) + x q (1 − xq ) F ( xq ) , (16) here F ( x ) := f a ( x, q ). Putting F ( x ) =: P M ≥ f M x M , by (16) we have(1 − q M ) f M = q M − ( q M + a + q M − a + q ) f M − + q M − (1 + q M − ) f M − − q M − f M − for all M ∈ Z (we consider f M = 0 for M < g M := q − ( M ) f M , we have(1 − q M ) g M = ( q M + a + q M − a + q ) g M − + ( q + q M ) g M − − q g M − . Putting G ( x ) := P M ≥ g M x M , we have(1 − xq )(1 − x q ) G ( x ) = (1 + xq − a )(1 + xq a ) G ( xq ) , and hence G ( x ) = ( − xq − a , − xq a ; q ) ∞ ( xq ; q ) ∞ ( x q ; q ) ∞ = ( − xq − a , − xq a ; q ) ∞ ( x q ; q ) ∞ , (17)where the latter equality is because a = 1 ,
2. By (A) and (B) (in §
2) we have G ( x ) = X i,j,k ≥ q ( i ) +(3 − a ) i ( q ; q ) i q ( j ) +(2+ a ) j ( q ; q ) j q k ( q ; q ) k x i + j +2 k . Finally, since f M := q ( M ) g M we have F ( x ) = X i,j,k ≥ q ( i ) +(3 − a ) i ( q ; q ) i q ( j ) +(2+ a ) j ( q ; q ) j q k ( q ; q ) k q ( i + j +2 k ) x i + j +2 k , which is precisely Theorem 7.3. (cid:3) Remark 7.4.
In place of (17) , if we write G ( x ) = ( − xq ; q ) ∞ ( x q ; q ) ∞ for a = 1 and G ( x ) = (1 + xq ) ( − xq ; q ) ∞ ( x q ; q ) ∞ for a = 2 , then we get the double sum expression (14) and an alternative one to (15) f ( x, q ) = X i,j ≥ q ( i ) +12 ( j ) +6 ij +3 i +6 j ( q ; q ) i ( q ; q ) j x i +2 j (1 + xq i +6 j ) . Remark 7.5.
We can reprove Theorem 7.1 using the equation (16) . If we put G ( x ) = P M ≥ g M x M := F ( x ) / ( x ; q ) ∞ , h M := g M / ( − q ; q ) M and H ( x ) := P M ≥ h M x M , by a similar argument to the proof of Proposition 7.3, we get (1 − x )(1 − xq ) H ( x ) = (1 + xq − a )(1 + xq a ) H ( xq ) and H ( x ) = ( − xq − a , − xq a ; q ) ∞ / ( x ; q ) ∞ . Again, by similar arguments (using(A) and (B)), we see g M = X i + j + k = M q ; q ) i q ( j ) +(3 − a ) j ( q ; q ) j q ( k ) +(3+ a ) k ( q ; q ) k ( − q ; q ) M . Since F ( x ) = G ( x )( x ; q ) ∞ , by Appell’s Comparison Theorem [10, page 101] we get F (1) = ( q ; q ) ∞ lim M →∞ g M = ( − q ; q ) ∞ X j,k ≥ q ( j ) +(3 − a ) j ( q ; q ) j q ( k ) +(3+ a ) k ( q ; q ) k = ( − q ; q ) ∞ ( − q − a , − q a ; q ) ∞ , which proves Theorem 7.1. eferences [1] G.E. Andrews, The theory of partitions , Encyclopedia of Mathematics and its Applications,vol.2, Addison-Wesley, 1976.[2] G.E. Andrews,
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E-mail address : [email protected] Department of Mathematical and Computing Sciences, Tokyo Institute of Technol-ogy, Tokyo 152-8551, Japan
E-mail address : [email protected]@kurims.kyoto-u.ac.jp